This reminds me of pokémon games, where you can deposit part of your money, because if you lose a combat you also lose half of the carried money. That idea could be used in a Zelda-like game too, to force the player to use the bank. Or maybe you can use a small wallet for the hero, which forces him to deposit money if he has too much... In other contexts a bank could be pointless.

That's somewhat clever, reminds me of the Majora's Mask bank system, that's something I didn't added in my project yet, but looks like you gave me this idea

Majora's Mask bank system is the game that gave me the idea for making my first bank script in Rpg Maker VX Ace.

So far the max withdraw/deposit and input is 9999, but I plan to have it at 999,999 at some point. That way, maybe the hero can buy houses and provide or create villages with the money. The gold in the bank could gain interest with a timer or the player might not be able to withdraw due to the economy or the village being under attack. The money in the bank could be a living condition for a job in the game because the player might need to be in clean clothing. Also, as Diarandor mentioned magic fairies can take part of the gold to the bank after a battle victory.

I have always wondered how money transported to the mother in the second generation Pokemon games. Sadly, they never implemented the system again after the second generation games. The last one I played was Pokémon Platinum. Since then, nothing close to a Pokemon bank as far as I know.